A New Path-Integral Representation of the $T$-Matrix in Potential Scattering

TL;DR

This paper introduces a new path-integral formulation for the $T$-matrix in potential scattering, simplifying calculations and enabling effective variational approximations, with accurate results at high energies and a novel low-energy scattering length representation.

Contribution

It presents a simplified path-integral representation of the $T$-matrix that avoids fictitious variables, facilitating variational methods in scattering analysis.

Findings

Accurate high-energy scattering results with Gaussian potentials.

New path-integral expression for low-energy scattering length.

Good agreement with exact partial-wave calculations.

Abstract

We employ the method used by Barbashov and collaborators in Quantum Field Theory to derive a path-integral representation of the $T$-matrix in nonrelativistic potential scattering which is free of functional integration over fictitious variables as was necessary before. The resulting expression serves as a starting point for a variational approximation applied to high-energy scattering from a Gaussian potential. Good agreement with exact partial-wave calculations is found even at large scattering angles. A novel path-integral representation of the scattering length is obtained in the low-energy limit.